Deformed integrable models from holomorphic Chern-Simons theory

We study the approaches to two-dimensional integrable field theories via a six-dimensional(6D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces four-dimensional theory, while under solving along fibres leads four-dimensional(4D) anti-self-dual Yang-Mills or its generalizations. From both theories, various can be obtained. In this work, we try investigate several deformations in framework. find that $\lambda$-deformation, rational $\eta$-deformation and generalized $\lambda$-deformation not realized from 4D model approach, even though they could obtained theory. The obstacle stems incompatibility between reduction boundary conditions. Nevertheless, show coupled of trigonometric description 6D ways, by considering case $(3,0)$-form is allowed have zeros.